Chaos in the Quasiperiodically Forced Helmholtz-Duffing Oscillator With Two-Well Potential

2005 ◽  
Author(s):  
Jing-Jun Lou ◽  
Shi-Jian Zhu ◽  
Qi-Wei He
Keyword(s):  
Hamiltonian System ◽  
Parameter Space ◽  
Homoclinic Orbit ◽  
Chaotic Dynamics ◽  
Chaotic Behavior ◽  
Duffing Oscillator ◽  
Restoring Force ◽  
Melnikov's Method ◽  
Melnikov’S Method ◽  
The Asymmetry

The chaotic dynamics of the quasiperiodically excited Helmholtz-Duffing oscillator with two-well potential was investigated. The condition of the existence of homoclinic orbit in the corresponding Hamiltonian system was presented which is asymmetrical resulting from the asymmetry restoring force. It was found that the mechanism for chaos is transverse homoclinic tori and it is illustrated how transverse homoclinic tori give rise to chaos for the Helmholtz-Duffing oscillator with multi-frequency periodic forces. The criterion for the existence of chaos was given utilizing a generalization of the Melnikov’s method. The region in parameter space where chaotic dynamics may occur was given. It was also demonstrated that increasing the number of forcing frequencies increases the area in parameter space where chaotic behavior can occur.

scholarly journals A computational framework for finding parameter sets associated with chaotic dynamics

2021 ◽  
pp. 1-11
Author(s):  
S. Koshy-Chenthittayil ◽  
E. Dimitrova ◽  
E.W. Jenkins ◽  
B.C. Dean
Keyword(s):  
Experimental Data ◽  
Dynamical Systems ◽  
Lyapunov Exponents ◽  
Parameter Space ◽  
Chaotic Dynamics ◽  
Chaotic Behavior ◽  
Computational Framework ◽  
Independent Variables ◽  
Systems Model ◽  
Small Support

Many biological ecosystems exhibit chaotic behavior, demonstrated either analytically using parameter choices in an associated dynamical systems model or empirically through analysis of experimental data. In this paper, we use existing software tools (COPASI, R) to explore dynamical systems and uncover regions with positive Lyapunov exponents where thus chaos exists. We evaluate the ability of the software’s optimization algorithms to find these positive values with several dynamical systems used to model biological populations. The algorithms have been able to identify parameter sets which lead to positive Lyapunov exponents, even when those exponents lie in regions with small support. For one of the examined systems, we observed that positive Lyapunov exponents were not uncovered when executing a search over the parameter space with small spacings between values of the independent variables.

Heteroclinic Bifurcations in Rigid Bodies Containing Internally Moving Parts and a Viscous Damper

1999 ◽  
Vol 66 (3) ◽  
pp. 720-728 ◽  
Author(s):  
G. L. Gray ◽  
D. C. Kammer ◽  
I. Dobson ◽  
A. J. Miller
Keyword(s):  
Chaotic Dynamics ◽  
Equations Of Motion ◽  
Major Axis ◽  
Rigid Bodies ◽  
Necessary Condition ◽  
Viscous Damping ◽  
Melnikov's Method ◽  
Melnikov’S Method ◽  
Melnikov Criterion ◽  
Analytical Criterion

Melnikov’s method is used to analytically study chaotic dynamics in an attitude transition maneuver of a torque-free rigid body in going from minor axis to major axis spin under the influence of viscous damping and nonautonomous perturbations. The equations of motion are presented, their phase space is discussed, and then they are transformed into a form suitable for the application of Melnikov’s method. Melnikov’s method yields an analytical criterion for homoclinic chaos in the form of an inequality that gives a necessary condition for chaotic dynamics in terms of the system parameters. The criterion is evaluated for its physical significance and for its application to the design of spacecraft. In addition, the Melnikov criterion is compared with numerical simulations of the system. The dependence of the onset of chaos on quantities such as body shape and magnitude of damping are investigated. In particular, it is found that for certain ranges of viscous damping values, the rate of kinetic energy dissipation goes down when damping is increased. This has a profound effect on the criterion for chaos.

scholarly journals Standing Swells Surveyed Showing Surprisingly Stable Solutions for the Lorenz '96 Model

2014 ◽  
Vol 24 (10) ◽  
pp. 1430027 ◽  
Author(s):  
Morgan R. Frank ◽  
Lewis Mitchell ◽  
Peter Sheridan Dodds ◽  
Christopher M. Danforth
Keyword(s):  
Parameter Space ◽  
Chaotic Dynamics ◽  
Degrees Of Freedom ◽  
Chaotic Behavior ◽  
Dynamical Behavior ◽  
Standing Waves ◽  
Nonlinear Dynamical ◽  
Hidden Order ◽  
Stable Solutions ◽  
Lorenz 96

The Lorenz '96 model is an adjustable dimension system of ODEs exhibiting chaotic behavior representative of the dynamics observed in the Earth's atmosphere. In the present study, we characterize statistical properties of the chaotic dynamics while varying the degrees of freedom and the forcing. Tuning the dimensionality of the system, we find regions of parameter space with surprising stability in the form of standing waves traveling amongst the slow oscillators. The boundaries of these stable regions fluctuate regularly with the number of slow oscillators. These results demonstrate hidden order in the Lorenz '96 system, strengthening the evidence for its role as a hallmark representative of nonlinear dynamical behavior.

Bifurcations and Chaotic Motions of a Class of Mechanical System With Parametric Excitations

2015 ◽  
Vol 10 (5) ◽  
Author(s):  
Liangqiang Zhou ◽  
Fangqi Chen ◽  
Yushu Chen
Keyword(s):  
Mechanical System ◽  
Parametric Excitation ◽  
Chaotic Dynamics ◽  
Stable And Unstable Manifolds ◽  
Chaotic Motions ◽  
Melnikov's Method ◽  
Melnikov’S Method ◽  
Critical Curves ◽  
Unstable Manifolds

Bifurcations and chaotic motions of a class of mechanical system subjected to a superharmonic parametric excitation or a nonlinear periodic parametric excitation are studied, respectively, in this paper. Chaos arising from the transverse intersections of the stable and unstable manifolds of the homoclinic and heteroclincic orbits is analyzed by Melnikov's method. The critical curves separating the chaotic and nonchaotic regions are plotted. Chaotic dynamics are compared for these systems with a periodic parametric excitation or a superharmonic parametric excitation, or a nonlinear periodic parametric excitation. Especially, some new dynamical phenomena are presented for the system with a nonlinear periodic parametric excitation.

QUANTIFYING SMOOTH AND NONSMOOTH REGULAR AND CHAOTIC DYNAMICS

2005 ◽  
Vol 15 (06) ◽  
pp. 2041-2055 ◽  
Author(s):  
J. AWREJCEWICZ ◽  
L. DZYUBAK
Keyword(s):  
Nonlinear System ◽  
Differential Equations ◽  
Numerical Method ◽  
Chaotic Dynamics ◽  
Lorenz System ◽  
Chaotic Behavior ◽  
Duffing Oscillator ◽  
Stick Slip ◽  
Excited System ◽  
Two Degree Of Freedom

This paper addresses two main paths of investigations. First, a new numerical method to trace regular and chaotic domains of any nonlinear system governed by ordinary differential equations is proposed. Second, the introduced approach is first testified using the well-known chaotic behavior of a Duffing oscillator and Lorenz system, and is then applied to analysis of discontinuous two-degree-of-freedom self-excited system with friction. Stick-slip and slip-slip chaos is reported, among others.

MELNIKOV'S METHOD AND STICK–SLIP CHAOTIC OSCILLATIONS IN VERY WEAKLY FORCED MECHANICAL SYSTEMS

1999 ◽  
Vol 09 (03) ◽  
pp. 505-518 ◽  
Author(s):  
J. AWREJCEWICZ ◽  
M. M. HOLICKE
Keyword(s):  
Numerical Simulation ◽  
Chaotic Dynamics ◽  
Mechanical Systems ◽  
Degree Of Freedom ◽  
Chaotic Oscillations ◽  
Stick Slip ◽  
Melnikov's Method ◽  
Melnikov’S Method

In this paper we predict stick–slip chaotic dynamics in a one-degree-of-freedom very weakly forced (quasiautonomous) oscillator using the Melnikov's technique. Numerical simulation confirms the validity of our approach.

Chaos in a Spacecraft Attitude Maneuver Due to Time-Periodic Perturbations

1996 ◽  
Vol 63 (2) ◽  
pp. 501-508 ◽  
Author(s):  
G. L. Gray ◽  
I. Dobson ◽  
D. C. Kammer
Keyword(s):  
Chaotic Motion ◽  
Chaotic Dynamics ◽  
Equations Of Motion ◽  
Major Axis ◽  
Artificial Satellites ◽  
Melnikov's Method ◽  
Melnikov’S Method ◽  
Attitude Maneuver ◽  
Analytical Criterion ◽  
Time Periodic

We use Melnikov’s method to study the chaotic dynamics of an attitude transition maneuver of a torque-free rigid body in going from minor axis spin to major axis spin under the influence of small damping. The chaotic motion is due to the formation of Smale horseshoes which are caused by the oscillation of small subbodies inside the satellite. The equations of motion are derived and then transformed into a form suitable for the application of Melnikov’s method. An analytical criterion for chaotic motion is derived in terms of the system parameters. This criterion is evaluated for its significance to the design of artificial satellites.

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